Carlo Rovelli - Collected Quotes

"Boltzmann has shown that entropy exists because we describe the world in a blurred fashion. He has demonstrated that entropy is precisely the quantity that counts how many are the different configurations that our blurred vision does not distinguish between. Heat, entropy, and the lower entropy of the past are notions that belong to an approximate, statistical description of nature. The difference between past and future is deeply linked to this blurring." (Carlo Rovelli, "The Order of Time", 2018)

"Continuity is only a mathematical technique for approximating very finely grained things. The world is subtly discrete, not continuous." (Carlo Rovelli, "The Order of Time", 2018)

"For a moving object, time contracts. Not only is there no single time for different places - there is not even a single time for any particular place. A duration can be associated only with the movement of something, with a given trajectory." (Carlo Rovelli, "The Order of Time", 2018)

"Granularity is ubiquitous in nature: light is made of photons, the particles of light. The energy of electrons in atoms can acquire only certain values and not others. The purest air is granular, and so, too, is the densest matter. Once it is understood that Newton’s space and time are physical entities like all others, it is natural to suppose that they are also granular. Theory confirms this idea: loop quantum gravity predicts that elementary temporal leaps are small, but finite." (Carlo Rovelli, "The Order of Time", 2018)

"If one section of the molecules is still, it becomes stirred up by the frenzy of neighboring ones that set them in motion, too: the agitation spreads, the molecules bump into and shove each other. In this way, cold things are heated in contact with hot ones: their molecules become jostled by hot ones and pushed into ferment. That is, they heat up." (Carlo Rovelli, "The Order of Time", 2018)

"In the elementary equations of the world, the arrow of time appears only where there is heat. The link between time and heat is therefore fundamental: every time a difference is manifested between the past and the future, heat is involved. In every sequence of events that becomes absurd if projected backward, there is something that is heating up." (Carlo Rovelli, "The Order of Time", 2018)

"It is not possible to think of duration as continuous. We must think of it as discontinuous: not as something that flows uniformly but as something that in a certain sense jumps, kangaroo-like, from one value to another. In other words, a minimum interval of time exists. Below this, the notion of time does not exist - even in its most basic meaning." (Carlo Rovelli, "The Order of Time", 2018)

"Nature, for its part, is what it is - and we discover it very gradually. If our grammar and our intuition do not readily adapt to what we discover, well, too bad: we must seek to adapt them." (Carlo Rovelli, "The Order of Time", 2018)

"Nothing is valid always and everywhere. Sooner or later, we always come across something that is new." (Carlo Rovelli, "The Order of Time", 2018)

"[...] our vision of the world is blurred because the physical interactions between the part of the world to which we belong and the rest are blind to many variables. This blurring is at the heart of Boltzmann's theory. From this blurring, the concepts of heat and entropy are born - and these are linked to the phenomena that characterize the flow of time. The entropy of a system depends explicitly on blurring. It depends on what I do not register, because it depends on the number of indistinguishable configurations. The same microscopic configuration may be of high entropy with regard to one blurring and low in relation to another." (Carlo Rovelli, "The Order of Time", 2018)

"Physics does not describe how things evolve 'in time' but how things evolve in their own times, and how 'times' evolve relative to each other." (Carlo Rovelli, "The Order of Time", 2018)

"Spacetime is a physical object like an electron. It, too, fluctuates. It, too, can be in a 'superposition' of different configurations." (Carlo Rovelli, "The Order of Time", 2018)

"The basic units in terms of which we comprehend the world are not located in some specific point in space. They are - if they are at all - in a where but also in a when. They are spatially but also temporally delimited: they are events." (Carlo Rovelli, "The Order of Time", 2018)

"The entire evolution of science would suggest that the best grammar for thinking about the world is that of change, not of permanence. Not of being, but of becoming." (Carlo Rovelli, "The Order of Time", 2018) 

"The entropy of the world does not depend only on the configuration of the world; it also depends on the way in which we are blurring the world, and this depends on what the variables of the world are that we interact with. That is to say, on the variables with which our part of the world interacts. [...] The difference between past and future does not lie in the elementary laws of motion; it does not reside in the deep grammar of nature. It is the natural disordering that leads to gradually less particular, less special situations."" (Carlo Rovelli, "The Order of Time", 2018)

"The entropy of the world in the far past appears very low to us. But this might not reflect the exact state of the world: it might regard the subset of the world's variables with which we, as physical systems, have interacted. It is with respect to the dramatic blurring produced by our interactions with the world, caused by the small set of macroscopic variables in terms of which we describe the world, that the entropy of the universe was low." (Carlo Rovelli, "The Order of Time", 2018)

"The relation of 'temporal precedence' is a partial order made of cones. Special relativity is the discovery that the temporal structure of the universe is like the one established by filiation: it defines an order between the events of the universe that is partial, not complete. The expanded present is the set of events that are neither past nor future: it exists, just as there are human beings who are neither our descendants nor our forebears. […] Every event has its past, its future, and a part of the universe that is neither past nor future, just as every person has forebears, descendants, and others who are neither forebears nor descendants. Light travels along the oblique lines that delimit these cones." (Carlo Rovelli, "The Order of Time", 2018)

"The world without a time variable is not a complicated one. It’s a net of interconnected events, where the variables in play adhere to probabilistic rules that, incredibly, we know for a good part how to write. And it’s a clear world, windswept and full of beauty as the crests of mountains; aridly beautiful as the cracked lips of the adolescent you loved." (Carlo Rovelli, "The Order of Time", 2018)

"Thermal agitation is like a continual shuffling of a pack of cards: if the cards are in order, the shuffling disorders them. In this way, heat passes from hot to cold, and not vice versa: by shuffling, by the natural disordering of everything. The growth of entropy is nothing other than the ubiquitous and familiar natural increase of disorder." (Carlo Rovelli, "The Order of Time", 2018)

"We cannot draw a complete map, a complete geometry, of everything that happens in the world, because such happenings - including among them the passage of time - are always triggered only by an interaction with, and with respect to, a physical system involved in the interaction. The world is like a collection of interrelated points of view. To speak of the world “seen from outside” makes no sense, because there is no “outside” to the world." (Carlo Rovelli, "The Order of Time", 2018)

"We often say that causes precede effects and yet, in the elementary grammar of things, there is no distinction between 'cause' and 'effect'. There are regularities, represented by what we call physical laws, that link events of different times, but they are symmetric between future and past. In a microscopic description, there can be no sense in which the past is different from the future."(Carlo Rovelli, "The Order of Time", 2018)

Menno-Jan Kraak - Collected Quotes

"Before a map can be drawn, a cartographer has to consider constraints that will influence its design. These include the purpose of the map, user characteristics, the use environment, and data characteristics. The purpose of a map, which can be manifold, relates to the questions the map will have to answer, or it use requirements. […] Above all, however, data characteristics will influence a mapmaker’s choice of symbology, because qualitative and quantitative data cannot be expressed in the same way." (Menno-Jan Kraak, "Mapping Time: Illustrated by Minard’s map of Napoleon’s Russian Campaign of 1812", 2014)

"Designs that play the role of presentation usually inform about spatial patterns and relations. This reflects the traditional cartographic approach. The cartographer begins with a known set of data and must select an appropriate visualization technique that will produce a high-quality explanation of facts. In other words, the visualization process ends with the best possible map. Map design, then, is important." (Menno-Jan Kraak, "Mapping Time: Illustrated by Minard’s map of Napoleon’s Russian Campaign of 1812", 2014)

"Graphic representations such as maps and diagrams should be able to answer all kinds of spatiotemporal questions, whatever the given situation. Not all graphics are suitable; even if they are, the data behind the graphics should be archived in a well-organized database." (Menno-Jan Kraak, "Mapping Time: Illustrated by Minard’s map of Napoleon’s Russian Campaign of 1812", 2014)

"In contrast to flow maps, origin-destination maps’ paths are highly structured, and do not use arrowheads to indicate direction. Both types of maps illustrate the volume of flow by varying the thickness of the path line’s shaft, some by gradually trimming the thickness of the shaft, others by splitting the shaft into sections and giving each section its own uniform thickness." (Menno-Jan Kraak, "Mapping Time: Illustrated by Minard’s map of Napoleon’s Russian Campaign of 1812", 2014)

"[…] information is lost or gained during the communication process. Information could be lost whenever a cartographer chooses to suppress information and/or users fail to understand all of it. It can be gained whenever the cartographer clarifies the original set of data and/or users combine the map information with their prior knowledge." (Menno-Jan Kraak, "Mapping Time: Illustrated by Minard’s map of Napoleon’s Russian Campaign of 1812", 2014)

"One cannot always be fully acquainted with the data, however. In such cases, exploration helps to judge the data’s usefulness. The exploration process, which emphasizes discovery, often facilitates an interactive, undirected search for structures and trends. This may result in conclusions that lead to alternative hypotheses." (Menno-Jan Kraak, "Mapping Time: Illustrated by Minard’s map of Napoleon’s Russian Campaign of 1812", 2014)

"We use maps to help us understand the world around us in the most effective and efficient way. Maps can summarize, clarify, explain, and emphasize aspects of our environment. Maps can play many roles. They support navigation and decision making, they of f er insight into spatial patterns and relationships among mapped phenomena, and […] they can tell stories. Maps do this well because they symbolize and abstract the reality they represent." (Menno-Jan Kraak, "Mapping Time: Illustrated by Minard’s map of Napoleon’s Russian Campaign of 1812", 2014)

Gerald J Whitrow - Collected Quotes

"In classical physics, most of the fundamental laws of nature were concerned either with the stability of certain configurations of bodies, e.g. the solar system, or else with the conservation of certain properties of matter, e.g. mass, energy, angular momentum or spin. The outstanding exception was the famous Second Law of Thermodynamics, discovered by Clausius in 1850. This law, as usually stated, refers to an abstract concept called entropy, which for any enclosed or thermally isolated system tends to increase continually with lapse of time. In practice, the most familiar example of this law occurs when two bodies are in contact: in general, heat tends to flow from the hotter body to the cooler. Thus, while the First Law of Thermodynamics, viz. the conservation of energy, is concerned only with time as mere duration, the Second Law involves the idea of trend." (Gerald J Whitrow, "The Structure of the Universe: An Introduction to Cosmology", 1949)

"Space-time is curved in the neighborhood of material masses, but it is not clear whether the presence of matter causes the curvature of space-time or whether this curvature is itself responsible for the existence of matter." (Gerald J Whitrow, "The Structure of the Universe: An Introduction to Cosmology", 1949)

"We have assumed that the laws of nature must be capable of expression in a form which is invariant for all possible transformations of the space-time co-ordinates." (Gerald J Whitrow, "The Structure of the Universe: An Introduction to Cosmology", 1949)

"The basic objection to attempts to deduce the unidirectional nature of time from concepts such as entropy is that they are attempts to reduce a more fundamental concept to a less fundamental one." (Gerald J Whitrow, "The Natural Philosophy of Time", 1961)

"[Time is not] a mysterious illusion of the intellect. [...] It is an essential feature of the universe." (Gerald J Whitrow, "The Natural Philosophy of Time", 1961)

"Our conscious appreciation of the fact that one event follows another is of a different kind from our awareness of either event separately. If two events are to be represented as occurring in succession, then - paradoxically - they must also be thought of simultaneously." (Gerald J Whitrow, "The Natural Philosophy of Time", 1961)

"Language itself inevitably introduced an element of permanence into the world. For, although speech itself is transitory, the conventionalized sound symbols of language transcended time." (Gerald J Whitrow, "Time in History: Views of Time from Prehistory to the Present Day", 1988)

"Man must have been conscious of memories and purposes long before he made any explicit distinction between past, present, and future." (Gerald J Whitrow, "Time in History: Views of Time from Prehistory to the Present Day", 1988)

"The development of rational thought actually seems to have impeded man's appreciation for the significance of time. [...] Belief that the ultimate reality is timeless is deeply rooted in human thinking, and the origin of rational investigation of the world was the search for permanent factors that lie behind the ever-changing pattern of events." (Gerald J Whitrow, "Time in History: Views of Time from Prehistory to the Present Day", 1988)

On Simultaneity III: Time

"Can each of them have a different time, and must there be more than one time running simultaneously? Surely not. For a time that is both equal and simultaneous is one and the same time, and even those that are not simultaneous are one in kind; for if there were dogs and horses, and seven of each, the number would be the same." (Aristotle, "Physics", 4th century BC)

"Further, if simultaneity in time, and not being before or after, means coinciding and being in the very ‘now’ wherein they coincide, then." (Aristotle, "Physics", 4th century BC)

"It is established that every continuum has further parts, and not so many parts finite in number that there are not further parts, and has all its parts actually and simultaneously, and therefore every continuum has simultaneously and actually infinitely many parts." (Gregory of Rimini [Gregorii Ariminensis], "Lectura super primum et secundum sententiarum", cca. 1350)

"There is a very different relationship between [...] space and duration. For we do not ascribe various durations to the different parts of space, but say that all endure together. The moment of duration is the same at Rome and at London, on the Earth and on the stars, and throughout the heavens. And just as we understand any moment of duration to be diffused throughout all spaces, according to its kind, without any thought of its parts, so it is no more contradictory that Mind also, according to its kind, can be diffused through space without any thought of its parts." (Isaac Newton, De Gravitatione et Aequipondio Fluridorum, cca. 1664-1668) 

"Just as space existed before the world was created and even now there exists an infinite space beyond the world (with which God coexists) [...] so time exists before the world and simultaneously with the world (prius mundo et simul cum mundo)." (Isaac Barrow, "Lectiones Geometricae", 1672)

"Not at all as far as its absolute, intrinsic nature is concerned. [,,,] Whether things run or stand still, whether we sleep or wake, time flows in its even tenor (aequo tenore tempus labitur). Even if all the stars would have remained at the places where they had been created, nothing would have been lost to time (nihil inde quicquam tempori decessisset). The temporal relations of earlier, afterwards, and simultaneity, even in that tranquil state, would have had their proper existence (prius, posterius, simul etiam in illo transquillo statu fuisset in se)."  (Isaac Barrow, "Lectiones Geometricae", 1672)

"Time absolutely is quantity, admitting in some manner the chief affections of quantity, equality, inequality, and proportion; nor do I believe there is anyone but allows that those things existed equal times, which rose and perished simultaneously." (Isaac Barrow, "Lectiones Geometricae", 1672)

"If a plurality of states of things is assumed to exist which involves no opposition to each other, they are said to exist simultaneously. Thus we deny that what occurred last year and this year are simultaneous, for they involve incompatible states of the same thing.  If one of two states which are not simultaneous involves a reason for the other, the former is held to be prior, the latter posterior. My earlier state involves a reason for the existence of my later state. And since my prior state, by reason of the connection between all things, involves the prior state of other things as well, it also involves a reason for the later state of these other things and is thus prior to them. Therefore whatever exists is either simultaneous with other existences or prior or posterior." (Gottfried W Leibniz, "Initium rerum Mathematicarum metaphysica", 1715)

"Time is the order of existence of those things which are not simultaneous. Thus time is the universal order of changes when we do not take into consideration the particular kinds of change. Duration is magnitude of time. If the magnitude of time is diminished uniformly and continuously, time disappears into moment, whose magnitude is zero. Space is the order of coexisting things, or the order of existence for things which are simultaneous." (Gottfried W Leibniz, "Initium rerum Mathematicarum metaphysica", 1715)

"In the same way, this should also happen with regard to time, namely, that between a preceding continuous time & the next following there should be a single instant, which is the indivisible boundary of either. There cannot be two instants, as we intimated above, contiguous to one another; but between one instant & another there must always intervene some interval of continuous time divisible indefinitely. In the same way, in any quantity which lasts for a continuous interval of time, there must be obtained a series of magnitudes of such a kind that to each instant of time there is its corresponding magnitude; & this magnitude connects the one that precedes with the one that follows it, & differs from the former by some definite magnitude. Nay even in that class of quantities, in which we cannot have two magnitudes at the same time, this very point can be deduced far more clearly, namely, that there cannot be any sudden change from one to another. For at that instant, when the sudden change should take place, & the series be broken by some momentary definite addition, two -magnitudes would necessarily be obtained, namely, the last of the first series & the first of the next. Now this very point is still more clearly seen in those states of things, in which on the one hand there must be at any instant some state so that at no time can the thing be without some state of the kind, whilst on the other hand it can never have two states of the kind simultaneously." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium, 1758)

"The difficulty of bringing idea and experience into relation with one another makes itself very painfully felt in all investigation of nature. The idea is independent of space and time. Research is limited in space and time. Hence in the idea simultaneous and successive features are most intimately linked, whereas these are always separated in experience; and to think of a process of nature as simultaneous and successive at once, in accordance with the idea, makes our heads spin. The understanding is unable to conceive of those sense data as jointly present which experience transmitted to it one at a time. Thus, the contradiction between ideation and perception remains forever unresolved." (Johann Wolfgang von Goethe, "Doubt and Resignation", 1820)

"Men have very good means of knowing in some cases, and of imagining in other cases, the distance between the points of space simultaneously occupied by the centres of two balls; if, at least, we be content to waive the difficulty as to imperfection of our means of ascertaining or specifying, or clearly idealising, simultaneity at distant places. For this we do commonly use signals by sound, by light, by electricity, by connecting wires or bars, or by various other means. The time required in the transmission of the signal involves an imperfection in human powers of ascertaining simultaneity of occurrences in distant places. It seems, however, probably not to involve any difficulty of idealising or imagining the existence of simultaneity." (James Thomson, "On the law of inertia, the principle of chronometry and the principle of absolute clinural rest, and of absolute rotation",” Proceedings of the Royal Society of Edinburgh 12, 1884)

"Time is the supreme illusion. It is but the inner prism by which we decompose being and life, the mode under which we perceive successively what is simultaneous in idea." (Henri-Frédéric Amiel, [journal entry] 1893)

"[...] if another way of measuring time would be adopted, the experiments on which Newton’s law is founded would none the less have the same meaning. Only the enunciation of the law would be different, because it would be translated into another language. [...] Time should be so defined that the equations of mechanics may be as simple as possible. In other words, there is not one way of measuring time more true than another; that which is generally adopted is only more convenient. Of two watches, we have no right to say that the one goes true, the other wrong; we can only say that it is advantageous to conform to the indications of the first." (Henri Poincaré, "La mesure du temps", 1898)

"I designate the following duplex thesis: any general attributive term - such as  the adjective ‘simultaneous’ and the abstract noun ‘simultaneity’ - (1) has ‘meaning’ only if its definition formulates some practicable method by which the applicability of the term in question to a given subject of discourse can be experimentally determined, i.e., describes some event capable of being directly observed at first hand under exactly determinable conditions, which event shall serve as the criterion for such applicability; and (2) the occurrence of such event, under the conditions set forth in the definition, is the meaning, and the whole meaning of the term." (Arthur O Lovejoy, 1930)

"[...] even in a temporal description of nature given by a relational theory of time. However, a theory, like the special theory of relativity, that denies the existence of an infinitely fast causal chain, deprives the concept of absolute simultaneity of its physical meaning even within a single inertial system. [...]  But since the metrical concept of velocity presupposes that we know the meaning of a transit time and since such a time, in turn, depends on a prior criterion of clock synchronization or simultaneity, we must first formulate the limiting property of electromagnetic chains [the fastest causal chain] without using the concept of simultaneity of noncoincident events." (Adolf Grünbaum, "Logical and philosophical foundations of the special theory of relativity", American Journal of Physics 23, 1955)

"The uncertainty principle refers to the degree of indeterminateness in the possible present knowledge of the simultaneous values of various quantities with which the quantum theory deals; it does not restrict, for example, the exactness of a position measurement alone or a velocity measurement alone." (Werner Heisenberg, "The Uncertainty Principle", [in James R Newman, "The World of Mathematics" Vol. II], 1956)

“The fundamental meaning of the Now is that of a universal simultaneity [...] it contains the whole world-wide extent of the simultaneous" (Eugen Fink, "Zur Ontologischen Frühgeschichte von Raum-Zeit-Bewegung", 1957)

"Synchronistic phenomena prove the simultaneous occurrence of meaningful equivalences in heterogenous, causally unrelated processes; in other words, they prove that a content perceived by an observer can, at the same time, be represented by an outside event, without any causal connection. From this it follows either that the psyche cannot be localized in time, or that space is relative to the psyche." (Carl G Jung, "The structure and dynamics of the psyche", 1960)

"The ‘relativity’ of the new theory - one of the most solidly verified theories in the entire range of physics - is chiefly, therefore, a relativity of simultaneity." (Ernan McMullin, "Simultaneity", 1967)

"According to the special theory there is a finite limit to the speed of causal chains, whereas classical causality allowed arbitrarily fast signals. Foundational studies […] soon revealed that this departure from classical causality in the special theory is intimately related to its most dramatic consequences: the relativity of simultaneity, time dilation, and length contraction. By now it had become clear that these kinematical effects are best seen as consequences of Minkowski space-time, which in turn incorporates a nonclassical theory of causal structure. However, it has not widely been recognized that the converse of this proposition is also true: the causal structure of Minkowski space-time contains within itself the entire geometry (topological and metrical structure) of Minkowski space-time." (John A. Winnie," The Causal Theory of Space-Time", 1977)

"It is hard to overestimate the impact of Einstein’s definition of distant simultaneity on philosophy in this century, set, as the words were, in the context of a highly successful theory of physics." (Graham Nerlich, "Simultaneity and convention in special relativity", 1982)

“If an event takes some time, while it happens, the now so to speak cuts through it, dividing that part of it which is already gone from that which is still to come. Two events which are thus cleaved by the same now are said to be simultaneous. Simultaneity, defined in this way, is evidently reflexive and symmetric, but it is not transitive. [...] However, if we conceive simultaneity as a relation between (idealized) durationless events we automatically ensure that it is transitive and hence an equivalence." (Roberto Torretti, Relativity and Geometry, 1983)

"There are many difficulties in application of [the games] theory to the real world. [...] In general, competitors are not in complete opposition. As a matter of fact often they don't even have the same objectives. This difficulty can often be circumvented by using a different objective, 'games of survival'. Secondly, a decision is seldom made once. This motivated the study of multistage games [...]. Finally, decisions are not usually made simultaneously. Recognition of this fact leads to 'games of protocol' [...]. Games of protocol can also be used to handle processes involving three or more people." (Richard E Bellman, "Eye of the Hurricane: An Autobiography", 1984)

"Much of what the universe had been, was, and would be, Newton had disclosed, was the outcome of an infinity of material particles all pulling on one another simultaneously. If the result of all that gravitational tussling had appeared to the Greeks to be a cosmos, it was simply because the underlying equation describing their behavior had itself turned out to be every bit a cosmos-orderly, beautiful, and decent." (Michael Guillen, "Five Equations That Changed the World", 1995)

"By means of a revision of the concept of simultaneity in a shapable form I arrived at the special relativity theory." (Albert Einstein)

Curved Spaces

"The integrals which we have obtained are not only general expressions which satisfy the differential equation, they represent in the most distinct manner the natural effect which is the object of the phenomenon [...] when this condition is fulfilled, the integral is, properly speaking, the equation of the phenomenon; it expresses clearly the character and progress of it, in the same manner as the finite equation of a line or curved surface makes known all the properties of those forms." (Jean-Baptiste-Joseph Fourier, "Théorie Analytique de la Chaleur", 1822)

"I hold: 1) that small portions of space are, in fact, of a nature analogous to little hills on a surface that is on the average fiat; namely, that the ordinary laws of geometry are not valid in them; 2) that this property of being curved or distorted is constantly being passed on from one portion of space to another after the manner of a wave; 3) that this variation of the curvature of space is what really happens in the phenomenon that we call the motion of matter, whether ponderable or ethereal; 4) that in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity." (William K Clifford, "On the Space Theory of Matter", [paper delivered before the Cambridge Philosophical Society, 1870)

[...] the time stream is curved helically in some higher dimension. In your case, a still further distortion brought two points of the coil into contact, and a sort of short circuit threw you into the higher curve. (Robert H Wilson, "A Flight Into Time", Wonder Stories, 1931)

"Any region of space-time that has no gravitating mass in its vicinity is uncurved, so that the geodesics here are straight lines, which means that particles move in straight courses at uniform speeds (Newton's first law). But the world-lines of planets, comets and terrestrial projectiles are geodesics in a region of space-time which is curved by the proximity of the sun or earth. […] No force of gravitation is […] needed to impress curvature on world-lines; the curvature is inherent in the space […]" (James H Jeans," The Growth of Physical Science", 1947) 

"Space-time is curved in the neighborhood of material masses, but it is not clear whether the presence of matter causes the curvature of space-time or whether this curvature is itself responsible for the existence of matter." (Gerald J Whitrow, "The Structure of the Universe: An Introduction to Cosmology", 1949)

"The mathematicians and physics men Have their mythology; they work alongside the truth, Never touching it; their equations are false But the things work. Or, when gross error appears, They invent new ones; they drop the theory of waves In universal ether and imagine curved space." (Robinson Jeffers," The Beginning and the End and Other Poems, The Great Wound", 1963)

"The ‘eyes of the mind’ must be able to see in the phase space of mechanics, in the space of elementary events of probability theory, in the curved four-dimensional space-time of general relativity, in the complex infinite dimensional projective space of quantum theory. To comprehend what is visible to the ‘actual eyes’, we must understand that it is only the projection of an infinite dimensional world on the retina." (Yuri I Manin, "Mathematics and Physics", 1981)

"Linking topology and dynamical systems is the possibility of using a shape to help visualize the whole range of behaviors of a system. For a simple system, the shape might be some kind of curved surface; for a complicated system, a manifold of many dimensions. A single point on such a surface represents the state of a system at an instant frozen in time. As a system progresses through time, the point moves, tracing an orbit across this surface. Bending the shape a little corresponds to changing the system's parameters, making a fluid more visous or driving a pendulum a little harder. Shapes that look roughly the same give roughly the same kinds of behavior. If you can visualize the shape, you can understand the system. (James Gleick, "Chaos: Making a New Science", 1987)

"Bodies like the earth are not made to move on curved orbits by a force called gravity; instead, they follow the nearest thing to a straight path in a curved space, which is called a geodesic. A geodesic is the shortest (or longest) path between two nearby points." (Stephen Hawking, "A Brief History of Time", 1988)

"Nonlinear systems (the graph of at least one relationship displays some curved feature) are notoriously more difficult to comprehend than linear systems, that is, they are more complex. Consequently they are also more difficult to control. This is exemplified by the volumes of elegant mathematics that have been developed in the search for optimal control of linear systems. (Robert L Flood & Ewart R Carson, "Dealing with Complexity: An introduction to the theory and application of systems", 1988)

"Linking topology and dynamical systems is the possibility of using a shape to help visualize the whole range of behaviors of a system. For a simple system, the shape might be some kind of curved surface; for a complicated system, a manifold of many dimensions. A single point on such a surface represents the state of a system at an instant frozen in time. As a system progresses through time, the point moves, tracing an orbit across this surface. Bending the shape a little corresponds to changing the system's parameters, making a fluid more visous or driving a pendulum a little harder. Shapes that look roughly the same give roughly the same kinds of behavior. If you can visualize the shape, you can understand the system." (James Gleick, "Chaos: Making a New Science", 1987)

"String theory promises to take a further step beyond that taken by Einstein's picture of force subsumed within curved space and time geometry. Indeed, string theory contains Einstein's theory of gravitation within itself. Loops of string behave like the exchange particles of the gravitational forces, or 'gravitons' as they are called in the point-particle picture of things. But it has been argued that it must be possible to extract even the geometry of space and time from the characteristics of the strings and their topological properties. At present, it is not known how to do this and we merely content ourselves with understanding how strings behave when they sit in a background universe of space and time." (John D. Barrow, "Theories of Everything: The Quest for Ultimate Explanation", 1991)

"[...] if we consider a topological space instead of a plane, then the question of whether the coordinates axes in that space are curved or straight becomes meaningless. The way we choose coordinate systems is related to the way we observe the property of smoothness in a topological space." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Spacetime tells matter how to move; matter tells spacetime how to curve." (John A Wheeler, "Geons, Black Holes and Quantum Foam: A Life in Physics" , 1998)

"General relativity explains gravitation as a curvature, or bending, or warping, of the geometry of space-time, produced by the presence of matter. Free fall in a space shuttle around Earth, where space is warped, produces weightlessness, and is equivalent from the observer's point of view to freely moving in empty space where there is no large massive body producing curvature. In free fall we move along a 'geodesic' in the curved space-time, which is essentially a straight-line motion over small distances. But it becomes a curved trajectory when viewed at large distances. This is what produces the closed elliptical orbits of planets, with tiny corrections that have been correctly predicted and measured. Planets in orbits are actually in free fall in a curved space-time!" (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"Apparent Impossibilities that Are New Truths […] irrational numbers, imaginary numbers, points at infinity, curved space, ideals, and various types of infinity. These ideas seem impossible at first because our intuition cannot grasp them, but they can be captured with the help of mathematical symbolism, which is a kind of technological extension of our senses." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"Mathematical language is littered with pejorative and mystical terms - such as irrational, imaginary, surd, transcendental - that were once used to ridicule supposedly impossible objects. And these are just terms applied to numbers. Geometry also has many concepts that seem impossible to most people, such as the fourth dimension, finite universes, and curved space - yet geometers (and physicists) cannot do without them. Thus there is no doubt that mathematics flirts with the impossible, and seems to make progress by doing so." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"We can describe general relativity using either of two mathematically equivalent ideas: curved space-time or metric field. Mathematicians, mystics and specialists in general relativity tend to like the geometric view because of its elegance. Physicists trained in the more empirical tradition of high-energy physics and quantum field theory tend to prefer the field view, because it corresponds better to how we (or our computers) do concrete calculations." (Frank Wilczek, "The Lightness of Being: Mass, Ether, and the Unification of Forces", 2008)

On Series III: Time Series

"Diagrams are sometimes used, not merely to convey several pieces of information such as several time series on one chart, but also to provide visual evidence of relationships between the series." (Alfred R Ilersic, "Statistics", 1959)

"Numerical data, which have been recorded at intervals of time, form what is generally described as a time series. [...] The purpose of analyzing time series is not always the determination of the trend by itself. Interest may be centered on the seasonal movement displayed by the series and, in such a case, the determination of the trend is merely a stage in the process of measuring and analyzing the seasonal variation. If a regular basic or under- lying seasonal movement can be clearly established, forecasting of future movements becomes rather less a matter of guesswork and more a matter of intelligent forecasting." (Alfred R Ilersic, "Statistics", 1959)

"Time series analysis often requires more knowledge of the data and relevant information about their background than it does of statistical techniques. Whereas the data in some other fields may be controlled so as to increase their representativeness, economic data are so changeable in their nature that it is usually impossible to sort out the separate effects of the various influences. Attempts to isolate cyclical, seasonal and irregular, or random movements, are made primarily in the hope that some underlying pattern of change over time may be revealed."  (Alfred R Ilersic, "Statistics", 1959)

"No observations are absolutely trustworthy. In no field of observation can we entirely rule out the possibility that an observation is vitiated by a large measurement or execution error. If a reading is found to lie a very long way from its fellows in a series of replicate observations, there must be a suspicion that the deviation is caused by a blunder or gross error of some kind. [...] One sufficiently erroneous reading can wreck the whole of a statistical analysis, however many observations there are." (Francis J Anscombe, "Rejection of Outliers", Technometrics Vol. 2 (2), 1960)

"A time series is a sequence of observations, usually ordered in time, although in some cases the ordering may be according to another dimension. The feature of time series analysis which distinguishes it from other statistical analysis is the explicit recognition of the importance of the order in which the observations are made. While in many problems the observations are statistically independent, in time series successive observations may be dependent, and the dependence may depend on the positions in the sequence. The nature of a series and the structure of its generating process also may involve in other ways the sequence in which the observations are taken." (Theodore W Anderson, "The Statistical Analysis of Time Series", 1971)

"Entropy theory, on the other hand, is not concerned with the probability of succession in a series of items but with the overall distribution of kinds of items in a given arrangement." (Rudolf Arnheim, "Entropy and Art: An Essay on Disorder and Order", 1974) 

"This transition from uncertainty to near certainty when we observe long series of events, or large systems, is an essential theme in the study of chance." (David Ruelle, "Chance and Chaos", 1991)

"System dynamics models are not derived statistically from time-series data. Instead, they are statements about system structure and the policies that guide decisions. Models contain the assumptions being made about a system. A model is only as good as the expertise which lies behind its formulation. A good computer model is distinguished from a poor one by the degree to which it captures the essence of a system that it represents. Many other kinds of mathematical models are limited because they will not accept the multiple-feedback-loop and nonlinear nature of real systems." (Jay W Forrester, "Counterintuitive Behavior of Social Systems", 1995)

"Like modeling, which involves making a static one-time prediction based on current information, time-series prediction involves looking at current information and predicting what is going to happen. However, with time-series predictions, we typically are looking at what has happened for some period back through time and predicting for some point in the future. The temporal or time element makes time-series prediction both more difficult and more rewarding. Someone who can predict the future based on what has occurred in the past can clearly have tremendous advantages over someone who cannot." (Joseph P Bigus,"Data Mining with Neural Networks: Solving business problems from application development to decision support", 1996)

"Many of the basic functions performed by neural networks are mirrored by human abilities. These include making distinctions between items (classification), dividing similar things into groups (clustering), associating two or more things (associative memory), learning to predict outcomes based on examples (modeling), being able to predict into the future (time-series forecasting), and finally juggling multiple goals and coming up with a good-enough solution (constraint satisfaction)." (Joseph P Bigus,"Data Mining with Neural Networks: Solving business problems from application development to decision support", 1996)

"Averages, ranges, and histograms all obscure the time-order for the data. If the time-order for the data shows some sort of definite pattern, then the obscuring of this pattern by the use of averages, ranges, or histograms can mislead the user. Since all data occur in time, virtually all data will have a time-order. In some cases this time-order is the essential context which must be preserved in the presentation." (Donald J Wheeler," Understanding Variation: The Key to Managing Chaos" 2nd Ed., 2000)

"No comparison between two values can be global. A simple comparison between the current figure and some previous value and convey the behavior of any time series. […] While it is simple and easy to compare one number with another number, such comparisons are limited and weak. They are limited because of the amount of data used, and they are weak because both of the numbers are subject to the variation that is inevitably present in weak world data. Since both the current value and the earlier value are subject to this variation, it will always be difficult to determine just how much of the difference between the values is due to variation in the numbers, and how much, if any, of the difference is due to real changes in the process." (Donald J Wheeler, "Understanding Variation: The Key to Managing Chaos" 2nd Ed., 2000)

"Prior to the discovery of the butterfly effect it was generally believed that small differences averaged out and were of no real significance. The butterfly effect showed that small things do matter. This has major implications for our notions of predictability, as over time these small differences can lead to quite unpredictable outcomes. For example, first of all, can we be sure that we are aware of all the small things that affect any given system or situation? Second, how do we know how these will affect the long-term outcome of the system or situation under study? The butterfly effect demonstrates the near impossibility of determining with any real degree of accuracy the long term outcomes of a series of events." (Elizabeth McMillan, Complexity, "Management and the Dynamics of Change: Challenges for practice", 2008)

"Regression toward the mean. That is, in any series of random events an extraordinary event is most likely to be followed, due purely to chance, by a more ordinary one." (Leonard Mlodinow, "The Drunkard’s Walk: How Randomness Rules Our Lives", 2008)

"Using random processes in our models allows economists to capture the variability of time series data, but it also poses challenges to model builders. As model builders, we must understand the uncertainty from two different perspectives. Consider first that of the econometrician, standing outside an economic model, who must assess its congruence with reality, inclusive of its random perturbations. An econometrician’s role is to choose among different parameters that together describe a family of possible models to best mimic measured real world time series and to test the implications of these models. I refer to this as outside uncertainty. Second, agents inside our model, be it consumers, entrepreneurs, or policy makers, must also confront uncertainty as they make decisions. I refer to this as inside uncertainty, as it pertains to the decision-makers within the model. What do these agents know? From what information can they learn? With how much confidence do they forecast the future? The modeler’s choice regarding insiders’ perspectives on an uncertain future can have significant consequences for each model’s equilibrium outcomes." (Lars P Hansen, "Uncertainty Outside and Inside Economic Models", [Nobel lecture] 2013)

"With time series though, there is absolutely no substitute for plotting. The pertinent pattern might end up being a sharp spike followed by a gentle taper down. Or, maybe there are weird plateaus. There could be noisy spikes that have to be filtered out. A good way to look at it is this: means and standard deviations are based on the naïve assumption that data follows pretty bell curves, but there is no corresponding 'default' assumption for time series data (at least, not one that works well with any frequency), so you always have to look at the data to get a sense of what’s normal. [...] Along the lines of figuring out what patterns to expect, when you are exploring time series data, it is immensely useful to be able to zoom in and out." (Field Cady, "The Data Science Handbook", 2017)

"[Making reasoned macro calls] starts with having the best and longest-time-series data you can find. You may have to take some risks in terms of the quality of data sources, but it amazes me how people are often more willing to act based on little or no data than to use data that is a challenge to assemble." (Robert J Shiller)

On Coincidence I

"It is no great wonder if in long process of time, while fortune takes her course hither and thither, numerous coincidences should spontaneously occur. If the number and variety of subjects to be wrought upon be infinite, it is all the more easy for fortune, with such an abundance of material, to effect this similarity of results." (Plutarch, Life of Sertorius, 1st century BC)

"We have here spoken of the prediction of facts of the same kind as those from which our rule was collected. But the evidence in favour of our induction is of a much higher and more forcible character when it enables us to explain and determine cases of a kind different from those which were contemplated in the formation of our hypothesis. The instances in which this has occurred, indeed, impress us with a conviction that the truth of our hypothesis is certain. No accident could give rise to such an extraordinary coincidence. No false supposition could, after being adjusted to one class of phenomena, so exactly represent a different class, when the agreement was unforeseen and contemplated. That rules springing from remote and unconnected quarters should thus leap to the same point, can only arise from that being where truth resides." (William Whewell, "The Philosophy of the Inductive Sciences" Vol. 2, 1840)

"Coincidences, in general, are great stumbling blocks in the way of that class of thinkers who have been educated to know nothing of the theory of probabilities - that theory to which the most glorious objects of human research are indebted for the most glorious of illustrations." (Edgar A Poe, "The Murders in the Rue Morgue", 1841)

"We produce these representations in and from ourselves with the same necessity with which the spider spins. If we are forced to comprehend all things only under these forms, then it ceases to be amazing that in all things we actually comprehend nothing but these forms. For they must all bear within themselves the laws of number, and it is precisely number which is most astonishing in things. All that conformity to law, which impresses us so much in the movement of the stars and in chemical processes, coincides at bottom with those properties which we bring to things. Thus it is we who impress ourselves in this way." (Friedrich Nietzsche, "On Truth and Lie in an Extra-Moral Sense", 1873)

"Nothing is more certain in scientific method than that approximate coincidence alone can be expected. In the measurement of continuous quantity perfect correspondence must be accidental, and should give rise to suspicion rather than to satisfaction." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"Before we can completely explain a phenomenon we require not only to find its true cause, its chief relations to other causes, and all the conditions which determine how the cause operates, and what its effect and amount of effect are, but also all the coincidences." (George Gore, "The Art of Scientific Discovery", 1878)

"As science progress, it becomes more and more difficult to fit in the new facts when they will not fit in spontaneously. The older theories depend upon the coincidences of so many numerical results which can not be attributed to chance. We should not separate what has been joined together." (Henri Poincaré, "The Ether and Matter", 1912)

"By the laws of statistics we could probably approximate just how unlikely it is that it would happen. But people forget - especially those who ought to know better, such as yourself - that while the laws of statistics tell you how unlikely a particular coincidence is, they state just as firmly that coincidences do happen." (Robert A Heinlein, "The Door Into Summer", 1957)

"It seems to me that it would be either a miracle or an unbelievable coincidence if all the major scientific theories […] somehow managed to co-operate with each other so as to conceal time’s arrow from us. There would be neither a miracle nor an unbelievable coincidence in the concealment of time’s arrow from us only if there were nothing to conceal - that is, if time had no arrows." (Henry Mehlberg) 

"Such properties seem to run through the fabric of the natural world like a thread of happy coincidences. But there are so many odd coincidences essential to life that some explanation seems required to account for them." (Sir Fred Hoyle)

Reality (From Fiction to Science-Fiction)

"The horror of the Same Old Thing is [...] an endless source of heresies in religion, folly in counsel, infidelity in marriage, and inconstancy in friendship. The humans live in time, and experience reality successively. To experience much of it, therefore, they must experience many different things; in other words, they must experience change. And since they need change, the Enemy (being a hedonist at heart) has made change pleasurable to them." (C. S. Lewis, "The Screwtape Letters", 1942)

"It is the normal lot of people who must live this life [in space] to be - by terrestrial standards—insane. Insanity under such conditions is a useful and logical defense mechanism, an invaluable and salutary retreat from reality." (Charles L Harness, "The Paradox Men", 1949)

"It seemed as if the structure of reality trembled for an instant, and that behind the world of the senses he caught a glimpse of another and totally different universe [...]" (Arthur C Clarke, "The City and the Stars", 1956)

"No live organism can continue for long to exist sanely under conditions of absolute reality; even larks and katydids are supposed, by some, to dream." (Shirley Jackson, "The Haunting of Hill House", 1959)

"Reality? It is only the illusion we can agree upon." (James Gunn, "The Joy Makers", 1961)

"When dreams become more important than reality, you give up travel, building, creating." (Gene Roddenberry, "Star Trek" ["The Menagerie"], 1966)

"The whole of modern so-called existence is an attempt to deny reality insofar as it exists."John Brunner, "Stand on Zanzibar", 1968)

"Reality, to me, is not so much something that you perceive, but something you make." (Philip K Dick, "The Android and the Human", 1972)

"The theory changes the reality it describes." (Philip K Dick, "Flow My Tears the Policeman Said", 1974)

"We exist in time. Time is what binds molecules to make your brown eyes, your yellow hair, your thick fingers. Time changes the structures, alters hair or fingers, dims the eyes, immutably mutating reality. Time, itself unchanging, is the cosmic glue, the universal antisolvent that holds our worlds together." (Marta Randall, "Secret Rider", 1976)

"There was no substitute for reality; one should beware of imitations." (Arthur C Clarke, "The Fountains of Paradise", 1979)

"Reality is that which when you stop believing in it, it doesn’t go away." (Philip K Dick, "Valis", 1981)

"The basic tool for the manipulation of reality is the manipulation of words. If you can control the meaning of words, you can control the people who must use the words." (Philip K Dick, "How to Build a Universe That Doesn’t Fall Apart Two Days Later", 1985)

"It is now known to science that there are many more dimensions than the classical four. Scientists say that these don’t normally impinge on the world because the extra dimensions are very small and curve in on themselves, and that since reality is fractal most of it is tucked inside itself. This means either that the universe is more full of wonders than we can hope to understand or, more probably, that scientists make things up as they go along." (Terry Pratchett, Pyramids, 1989)

"It is always hard when reality intrudes on belief." (Alan D Foster, "Cyber Way", 1990)

"The dreams of people are in the machines, a planet network of active imaginations hooked into their made-up, make-believe worlds. Artificial reality is taking over; it has its own children." (Storm Constantine, "Immaculate", 1991)

On Continuity VII (Spacetime)

"Time and space are divided into the same and equal divisions. Wherefore also, Zeno’s argument, that it is impossible to go through an infinite collection or to touch an infinite collection one by one in a finite time, is fallacious. For there are two senses in which the term ‘infinte’ is applied both to length and to time and in fact to all continuous things: either in regard to divisibility or in regard to number. Now it is not possible to touch things infinite as to number in a finite time, but it is possible to touch things infinite in regard to divisibility; for time itself is also infinite in this sense."  (Aristotle, "Physics", cca. 350 BC)

"Time with its continuity logically involves some other kind of continuity than its own. Time, as the universal form of change, cannot exist unless there is something to undergo change, and to undergo a change continuous in time, there must be a continuity of changeable qualities." (Charles S Peirce, "The Law of Mind", 1892)

"Motion consists merely in the occupation of different places at different times, subject to continuity. [...] There is no transition from place to place, no consecutive moment or consecutive position, no such thing as velocity except in the sense of a real number, which is the limit of a certain set of quotients." (Bertrand Russell's, "Principles of Mathematics", 1903)

"The discovery of Minkowski […] is to be found […] in the fact of his recognition that the four-dimensional space-time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space. In order to give due prominence to this relationship, however, we must replace the usual time co-ordinate t by an imaginary magnitude, v-1*ct, proportional to it. Under these conditions, the natural laws satisfying the demands of the (special) theory of relativity assume mathematical forms, in which the time co-ordinate plays exactly the same role as the three space-coordinates. Formally, these four co-ordinates correspond exactly to the three space co-ordinates in Euclidean geometry." (Albert Einstein,"Relativity: The Special and General Theory", 1920)

"The scene of action of reality is not a three-dimensional Euclidean space but rather a four-dimensional world, in which space and time are linked together indissolubly. However deep the chasm may be that separates the intuitive nature of space from that of time in our experience, nothing of this qualitative difference enters into the objective world which physics endeavors to crystallize out of direct experience. It is a four-dimensional continuum, which is neither 'time' nor 'space'. Only the consciousness that passes on in one portion of this world experiences the detached piece which comes to meet it and passes behind it as history, that is, as a process that is going forward in time and takes place in space." (Hermann Weyl, "Space, Time, Matter", 1922)

"In classical science, it was strange to find that action [...] should yet present the artificial aspect of an energy in space multiplied by a duration. As soon, however, as we realise that the fundamental continuum of the universe is one of space-time and not one of separate space and time, the reason for the importance of the seemingly artificial combination of space with time in the expression for the action receives a very simple explanation. Henceforth, action is no longer energy in a volume of space multiplied by a duration; it is simply energy in a volume of the world, that is to say, in a volume of four-dimensional space-time." (Aram D'Abro, "The Evolution of Scientific Thought from Newton to Einstein", 1927)

"[…] evolution is only one aspect of the order of nature, of the relations of cause and effect, of continuity of space and time, which pervade the universe and enable us to comprehend its simplicity of plan, its complexity of detail." (William D Matthew, Natural History Vol. 25 (2), 1925)

"[…] the universe is not a rigid and inimitable edifice where independent matter is housed in independent space and time; it is an amorphous continuum, without any fixed architecture, plastic and variable, constantly subject to change and distortion. Wherever there is matter and motion, the continuum is disturbed. Just as a fi sh swimming in the sea agitates the water around it, so a star, a comet, or a galaxy distorts the geometry of the spacetime through which it moves." (Lincoln Barnett, "The Universe and Dr. Einstein", 1948)

"The whole fabric of the space-time continuum is not merely curved, it is in fact totally bent." (Douglas N Adams, "The Restaurant at the End of the Universe", 1980)

"Time goes forward because energy itself is always moving from an available to an unavailable state. Our consciousness is continually recording the entropy change in the world around us. [...] we experience the passage of time by the succession of one event after another. And every time an event occurs anywhere in this world energy is expended and the overall entropy is increased. To say the world is running out of time then, to say the world is running out of usable energy. In the words of Sir Arthur Eddington, 'Entropy is time's arrow'." (Jeremy Rifkin, "Entropy", 1980)

"The view of science is that all processes ultimately run down, but entropy is maximized only in some far, far away future. The idea of entropy makes an assumption that the laws of the space-time continuum are infinitely and linearly extendable into the future. In the spiral time scheme of the timewave this assumption is not made. Rather, final time means passing out of one set of laws that are conditioning existence and into another radically different set of laws. The universe is seen as a series of compartmentalized eras or epochs whose laws are quite different from one another, with transitions from one epoch to another occurring with unexpected suddenness." (Terence McKenna, "True Hallucinations", 1989)

"Symmetry is bound up in many of the deepest patterns of Nature, and nowadays it is fundamental to our scientific understanding of the universe. Conservation principles, such as those for energy or momentum, express a symmetry that (we believe) is possessed by the entire space-time continuum: the laws of physics are the same everywhere." (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)

"Continuity is only a mathematical technique for approximating very finely grained things. The world is subtly discrete, not continuous." (Carlo Rovelli, "The Order of Time", 2018)

"It is not possible to think of duration as continuous. We must think of it as discontinuous: not as something that flows uniformly but as something that in a certain sense jumps, kangaroo-like, from one value to another. In other words, a minimum interval of time exists. Below this, the notion of time does not exist - even in its most basic meaning." (Carlo Rovelli, "The Order of Time", 2018)

"Space. The continual becoming: invisible fountain from which all rhythms flow and to which they must pass. Beyond time or infinity." (Frank L Wright)

On Manifolds V (Geometry III)

"Whereas the conception of space and time as a four-dimensional manifold has been very fruitful for mathematical physicists, its effect in the field of epistemology has been only to confuse the issue. Calling time the fourth dimension gives it an air of mystery. One might think that time can now be conceived as a kind of space and try in vain to add visually a fourth dimension to the three dimensions of space. It is essential to guard against such a misunderstanding of mathematical concepts. If we add time to space as a fourth dimension it does not lose any of its peculiar character as time. [...] Musical tones can be ordered according to volume and pitch and are thus brought into a two dimensional manifold. Similarly colors can be determined by the three basic colors red, green and blue… Such an ordering does not change either tones or colors; it is merely a mathematical expression of something that we have known and visualized for a long time. Our schematization of time as a fourth dimension therefore does not imply any changes in the conception of time. [...] the space of visualization is only one of many possible forms that add content to the conceptual frame. We would therefore not call the representation of the tone manifold by a plane the visual representation of the two dimensional tone manifold." (Hans Reichenbach, "The Philosophy of Space and Time", 1928)

"The sequence of numbers which grows beyond any stage already reached by passing to the next number is a manifold of possibilities open towards infinity, it remains forever in the status of creation, but is not a closed realm of things existing in themselves. That we blindly converted one into the other is the true source of our difficulties […]" (Hermann Weyl, "Mathematics and Logic", 1946)

"The first attempts to consider the behavior of so-called 'random neural nets' in a systematic way have led to a series of problems concerned with relations between the 'structure' and the 'function' of such nets. The 'structure' of a random net is not a clearly defined topological manifold such as could be used to describe a circuit with explicitly given connections. In a random neural net, one does not speak of 'this' neuron synapsing on 'that' one, but rather in terms of tendencies and probabilities associated with points or regions in the net." (Anatol Rapoport, "Cycle distributions in random nets", The Bulletin of Mathematical Biophysics 10(3), 1948)

"The main object of study in differential geometry is, at least for the moment, the differential manifolds, structures on the manifolds (Riemannian, complex, or other), and their admissible mappings. On a manifold the coordinates are valid only locally and do not have a geometric meaning themselves." (Shiing-Shen Chern, "Differential geometry, its past and its future", 1970)

"[...] a manifold is a set M on which 'nearness' is introduced (a topological space), and this nearness can be described at each point in M by using coordinates. It also requires that in an overlapping region, where two coordinate systems intersect, the coordinate transformation is given by differentiable transition functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"It is commonly said that the study of manifolds is, in general, the study of the generalization of the concept of surfaces. To some extent, this is true. However, defining it that way can lead to overshadowing by 'figures' such as geometrical surfaces." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"One could also question whether we are looking for a single overarching mathematical structure or a combination of different complementary points of view. Does a fundamental theory of Nature have a global definition, or do we have to work with a series of local definitions, like the charts and maps of a manifold, that describe physics in various 'duality frames'. At present string theory is very much formulated in the last kind of way." (Robbert Dijkgraaf, "Mathematical Structures", 2005)

"Quantum physics, in particular particle and string theory, has proven to be a remarkable fruitful source of inspiration for new topological invariants of knots and manifolds. With hindsight this should perhaps not come as a complete surprise. Roughly one can say that quantum theory takes a geometric object (a manifold, a knot, a map) and associates to it a (complex) number, that represents the probability amplitude for a certain physical process represented by the object." (Robbert Dijkgraaf, "Mathematical Structures", 2005)

"The primary aspects of the theory of complex manifolds are the geometric structure itself, its topological structure, coordinate systems, etc., and holomorphic functions and mappings and their properties. Algebraic geometry over the complex number field uses polynomial and rational functions of complex variables as the primary tools, but the underlying topological structures are similar to those that appear in complex manifold theory, and the nature of singularities in both the analytic and algebraic settings is also structurally very similar." (Raymond O Wells Jr, "Differential and Complex Geometry: Origins, Abstractions and Embeddings", 2017)

"Therefore one has taken everywhere the opposite road, and each time one encounters manifolds of several dimensions in geometry, as in the doctrine of definite integrals in the theory of imaginary quantities, one takes spatial intuition as an aid. It is well known how one gets thus a real overview over the subject and how only thus are precisely the essential points emphasized." (Bernhard Riemann)

On Spacetime (From Fiction to Science-Fiction)

"Some people who talk about the Fourth Dimension do not know they mean it. It is only another way of looking at Time. There is no difference between Time and any of the three dimensions of Space except that our consciousness moves along it." (Herbert G Wells, "The Time Machine: An Invention", 1895)

"[...] the time stream is curved helically in some higher dimension. In your case, a still further distortion brought two points of the coil into contact, and a sort of short circuit threw you into the higher curve." (Robert H Wilson, "A Flight Into Time", Wonder Stories, 1931)

"Man has natural three-dimensional limits, and he also has four-dimensional ones, considering time as an extension. When he reaches those limits, he ceases to grow and mature, and forms rigidly within the mold of those limiting walls. It is stasis, which is retrogression unless all else stands still as well. A man who reaches his limits is tending toward subhumanity. Only when he becomes superhuman in time and space can immortality become practical." (Henry Kuttner & Catherine L Moore [aka Lewis Padgett], "Time Enough", 1946)

"Space and Time aren’t real, apart. And they aren’t really different. They fade one into the other all around us." (Jack Williamson, "The Legion of Space", 1947)

"There are and have been worlds and cultures without end, each nursing the proud illusion that it is unique in space and time. There have been men without number suffering from the same megalomania; men who imagined themselves unique, irreplaceable, irreproducible. There will be more [...] more plus infinity." (Alfred Bester, "The Demolished Man", 1953)

"There is a fifth dimension beyond that which is known to Man. It is a dimension as vast as space and as timeless as infinity. It is the middle ground between light and shadow, between science and superstition, and it lies between the pit of man’s fears and the summit of his knowledge. This is the dimension of imagination. It is an area which we call ... The Twilight Zone." (Rod Serling, "The Twilight Zone" [TV series] 1959)

"The present, as every schoolboy knows, is only the surface of the space-time sea, and a living spacewhale can dive beneath this surface and sojourn in times past, can return, if it so desires, to the primordial moment when the cosmos was born." (Robert F Young, "Starscape with Frieze of Dreams", 1970)

"An infinity of universes swim in superspace, all passing through their own cycles of birth and death; some are novel, others repetitious; some produce macrolife, others do not; still others are lifeless. In time, macrolife will attempt to reach out from its cycles to other space-time bubbles, perhaps even to past cycles, which leave their echoes in superspace, and might be reached. In all these ambitions, only the ultimate pattern of development is unknown, drawing macrolife toward some future transformation still beyond its view. There are times when the oldest macrolife senses that vaster intelligences are peering in at it from some great beyond [...]" (George Zebrowski, "Macrolife", 1979)

"The whole fabric of the space-time continuum is not merely curved, it is in fact totally bent." (Douglas N Adams, "The Restaurant at the End of the Universe", 1980)

"Time and space were themselves players, vast lands engulfing the figures, a weave of future and past. There was no riverrun of years. The abiding loops of causality ran both forward and back. The timescape rippled with waves, roiled and flexed, a great beast in the dark sea." (Gregory Benford, "Timescape", 1980)

"The dimension of the imagination is much more complex than those of time and space, which are very junior dimensions indeed." (Terry Pratchett, "The Colour of Magic", 1983)

"History too has an inertia. In the four dimensions of spacetime, particles (or events) have directionality; mathematicians, trying to show this, draw what they call 'world lines' on graphs. In human affairs, individual world lines form a thick tangle, curling out of the darkness of prehistory and stretching through time: a cable the size of Earth itself, spiraling round the sun on a long curved course. That cable of tangled world lines is history. Seeing where it has been, it is clear where it is going - it is a matter of simple extrapolation." (Kim S Robinson, "Red Mars", 1992)

On Spacetime (2000-2019)

"Neither space nor time has any existence outside the system of evolving relationships that comprises the universe. Physicists refer to this feature of general relativity as background independence." (Lee Smolin, "Three Roads to Quantum Gravity", 2000)

"The relational picture of space and time has implications that are as radical as those of natural selection, not only for science but for our perspective on who we are and how we came to exist in this evolving universe of relations." (Lee Smolin, "Three Roads to Quantum Gravity", 2000)

"Time is described only in terms of change in the network of relationships that describes space." (Lee Smolin, "Three Roads to Quantum Gravity", 2000)

"In string theory one studies strings moving in a fixed classical spacetime. […] what we call a background-dependent approach. […] One of the fundamental discoveries of Einstein is that there is no fixed background. The very geometry of space and time is a dynamical system that evolves in time. The experimental observations that energy leaks from binary pulsars in the form of gravitational waves - at the rate predicted by general relativity to the […] accuracy of eleven decimal place - tell us that there is no more a fixed background of spacetime geometry than there are fixed crystal spheres holding the planets up." (Lee Smolin, "Loop Quantum Gravity", The New Humanists: Science at the Edge, 2003)

"Spacetime […] turns out to be discrete, described by a structure called spin foam." (Lee Smolin, “The New Humanists: Science at the Edge”, 2003)

"General relativity explains gravitation as a curvature, or bending, or warping, of the geometry of space-time, produced by the presence of matter. Free fall in a space shuttle around Earth, where space is warped, produces weightlessness, and is equivalent from the observer's point of view to freely moving in empty space where there is no large massive body producing curvature. In free fall we move along a 'geodesic' in the curved space-time, which is essentially a straight-line motion over small distances. But it becomes a curved trajectory when viewed at large distances. This is what produces the closed elliptical orbits of planets, with tiny corrections that have been correctly predicted and measured. Planets in orbits are actually in free fall in a curved space-time!" (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"Space and time capture the imagination like no other scientific subject. For good reason. They form the arena of reality, the very fabric of the cosmos." (Brian Greene, "The Fabric of the Cosmos", 2004)

"The space and time of the universe that we humans inhabit contain symmetries. These are almost obvious yet subtle, even mysterious. Space and time form the stage upon which the dynamics - that is, the motion and interactions of the physical systems, atoms, atomic nuclei, protozoa, and people - are played out. The symmetries of space and time control the dynamics of the physical interactions of matter." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"Minkowski calls a spatial point existing at a temporal point a world point. These coordinates are now called 'space-time coordinates'. The collection of all imaginable value systems or the set of space-time coordinates Minkowski called the world. This is now called the manifold. The manifold is four-dimensional and each of its space-time points represents an event." (Friedel Weinert," The Scientist as Philosopher: Philosophical Consequences of Great Scientific Discoveries", 2005) 

"Mathematicians call the infinite curvature limit of spacetime a singularity. In this picture, then, the big bang emerges from a singularity. The best way to think about singularities is as boundaries or edges of spacetime. In this respect they are not, technically, part of spacetime itself." (Paul Davies," Cosmic Jackpot: Why Our Universe Is Just Right for Life", 2007) 

"We can describe general relativity using either of two mathematically equivalent ideas: curved space-time or metric field. Mathematicians, mystics and specialists in general relativity tend to like the geometric view because of its elegance. Physicists trained in the more empirical tradition of high-energy physics and quantum field theory tend to prefer the field view, because it corresponds better to how we (or our computers) do concrete calculations." (Frank Wilczek, "The Lightness of Being: Mass, Ether, and the Unification of Forces", 2008)

"The hypothesis underlying all approaches to the landscape is that there is a cosmological setting in which different regions or epochs of the universe can have different effective laws. This implies the existence of spacetime regions not directly observable […] These regions must either be in the past of our big bang, or far enough away from us to be causally unrelated." (Lee Smolin," A perspective on the landscape problem", 2012)

"One of the most crucial developments in theoretical physics was the move from theories dependent on fixed, non-dynamical background space-time structures to background-independent theories, in which the space-time structures themselves are dynamical entities. [...] Even today, many physicists and philosophers do not fully understand the significance of this development, let alone accept it in practice. One must assume that, in an empty region of space-time, the points have no inherent individuating properties - nor indeed are there any spatio-temporal relations between them - that do not depend on the presence of some metric tensor field. [...] Thus, general relativity became the first fully dynamical, background- independent space-time theory." (John Stachel, "The Hole Argument", 2014)

"For a moving object, time contracts. Not only is there no single time for different places - there is not even a single time for any particular place. A duration can be associated only with the movement of something, with a given trajectory." (Carlo Rovelli, "The Order of Time", 2018)

"Granularity is ubiquitous in nature: light is made of photons, the particles of light. The energy of electrons in atoms can acquire only certain values and not others. The purest air is granular, and so, too, is the densest matter. Once it is understood that Newton’s space and time are physical entities like all others, it is natural to suppose that they are also granular. Theory confirms this idea: loop quantum gravity predicts that elementary temporal leaps are small, but finite." (Carlo Rovelli, "The Order of Time", 2018

"Spacetime is a physical object like an electron. It, too, fluctuates. It, too, can be in a 'superposition' of different configurations." (Carlo Rovelli, "The Order of Time", 2018)

"The basic units in terms of which we comprehend the world are not located in some specific point in space. They are - if they are at all - in a where but also in a when. They are spatially but also temporally delimited: they are events." (Carlo Rovelli, "The Order of Time", 2018)

Hermann Minkowski - Collected Quotes

"By laying down the relativity postulate from the outset, sufficient means have been created for deducing henceforth the complete series of Laws of Mechanics from the principle of conservation of energy (and statements concerning the form of the energy) alone." (Hermann Minkowski, "The Fundamental Equations for Electromagnetic Processes in Moving Bodies", 1907)

"Integers are the fountainhead of all mathematics." (Hermann Minkowski, "Diophantische Approximationen: eine einfuhrung in die zahlen Theorie", 1907)

"The equations of Newton's mechanics exhibit a two-fold invariance. Their form remains unaltered, firstly, if we subject the underlying system of spatial coordinates to any arbitrary change of position ; secondly, if we change its state of motion, namely, by imparting to it any uniform translatory motion ; furthermore, the zero point of time is given no part to play. We are accustomed to look upon the axioms of geometry as finished with, when we feel ripe for the axioms of mechanics, and for that reason the two invariances are probably rarely mentioned in the same breath. Each of them by itself signifies, for the differential equations of mechanics, a certain group of transformations. The existence of the first group is looked upon as a fundamental characteristic of space. The second group is preferably treated with disdain, so that we with un-troubled minds may overcome the difficulty of never being able to decide, from physical phenomena, whether space, which is supposed to be stationary, may not be after all in a state of uniform translation. Thus the two groups, side by side, lead their lives entirely apart. Their utterly heterogeneous character may have discouraged any attempt to compound them. But it is precisely when they are compounded that the complete group, as a whole, gives us to think." (Hermann Minkowski, "Space and Time" ["Raum und Zeit"], [Address to the 80th Assembly of German Natural Scientists and Physicians] 1908)

"The objects of our perception invariably include places and times in combination. Nobody has ever noticed a place except at a time, or a time except at a place. But I still respect the dogma that both space and time have independent significance. A point of space at a point of time, that is a system of values x, y, z, t, I will call a world-point." (Hermann Minkowski, "Space and Time" ["Raum und Zeit"], [Address to the 80th Assembly of German Natural Scientists and Physicians] 1908)

"The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth, space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality." (Hermann Minkowski, "Space and Time" ["Raum und Zeit"], [Address to the 80th Assembly of German Natural Scientists and Physicians] 1908)

"The whole world appears resolved into such world-lines. And I should like to say beforehand that, according to my opinion, it would be possible for the physical laws to find their fullest expression as correlations of these world-lines." (Hermann Minkowski, "Space and Time" ["Raum und Zeit"], [Address to the 80th Assembly of German Natural Scientists and Physicians] 1908)

"The objects of our perception invariably include places and times in combination. Nobody has ever noticed a place except at a time, or a time except at a place. But I still respect the dogma that both space and time have independent significance. A point of space at a point of time, that is a system of values x, y, z, t, I will call a world-point." (Hermann Minkowski, "The Principle of Relativity: A Collection of Original Memoirs on the, Special and General Theory of Relativity, Space and Time", Nature vol. 113, 1924)

"The rigid electron is in my view a monster in relation to Maxwell's equations, whose innermost harmony is the principle of relativity [...] the rigid electron is no working hypothesis, but a working hindrance. Approaching Maxwell's equations with the concept of the rigid electron seems to me the same thing as going to a concert with your ears stopped up with cotton wool. We must admire the courage and the power of the school of the rigid electron which leaps across the widest mathematical hurdles with fabulous hypotheses, with the hope to land safely over there on experimental-physical ground." (Hermann Minkowski [in Arthur I Miller, "Albert Einstein's Special Theory of Relativity", 1981)

On Noise I

"Noise is the most impertinent of all forms of interruption. It is not only an interruption, but also a disruption of thought." (Arthur Schopenhauer, "Parerga and Paralipomena", 1851)

"Mathematics is the predominant science of our time; its conquests grow daily, though without noise; he who does not employ it for himself, will some day find it employed against himself." (Johann F Herbart, Werke, 1890)

"Life pushes its way through this fatalistically determined world like a river flowing upstream. It is a system of utterly improbable order, a message in a world of noise." (Joseph H Rush, "The Dawn of Life", 1957)

"Higher, directed forms of energy (e.g., mechanical, electric, chemical) are dissipated, that is, progressively converted into the lowest form of energy, i.e., undirected heat movement of molecules; chemical systems tend toward equilibria with maximum entropy; machines wear out owing to friction; in communication channels, information can only be lost by conversion of messages into noise but not vice versa, and so forth." (Ludwig von Bertalanffy, "Robots, Men and Minds", 1967)

"To adapt to a changing environment, the system needs a variety of stable states that is large enough to react to all perturbations but not so large as to make its evolution uncontrollably chaotic. The most adequate states are selected according to their fitness, either directly by the environment, or by subsystems that have adapted to the environment at an earlier stage. Formally, the basic mechanism underlying self-organization is the (often noise-driven) variation which explores different regions in the system’s state space until it enters an attractor. This precludes further variation outside the attractor, and thus restricts the freedom of the system’s components to behave independently. This is equivalent to the increase of coherence, or decrease of statistical entropy, that defines self-organization." (Francis Heylighen, "The Science Of Self-Organization And Adaptivity", 1970)

"Probability plays a central role in many fields, from quantum mechanics to information theory, and even older fields use probability now that the presence of 'nois' is officially admitted. The newer aspects of many fields start with the admission of uncertainty." (Richard Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"An essential element of dynamics systems is a positive feedback that self-enhances the initial deviation from the mean. The avalanche is proverbial. Cities grow since they attract more people, and in the universe, a local accumulation of dust may attract more dust, eventually leading to the birth of a star. Earlier or later, self-enhancing processes evoke an antagonistic reaction. A collapsing stock market stimulates the purchase of shares at a low price, thereby stabilizing the market. The increasing noise, dirt, crime and traffic jams may discourage people from moving into a big city." (Hans Meinhardt, "The Algorithmic Beauty of Sea Shells", 1995)

"Rather mathematicians like to look for patterns, and the primes probably offer the ultimate challenge. When you look at a list of them stretching off to infinity, they look chaotic, like weeds growing through an expanse of grass representing all numbers. For centuries mathematicians have striven to find rhyme and reason amongst this jumble. Is there any music that we can hear in this random noise? Is there a fast way to spot that a particular number is prime? Once you have one prime, how much further must you count before you find the next one on the list? These are the sort of questions that have tantalized generations." (Marcus du Sautoy, "The Music of the Primes", 1998)

"Data are collected as a basis for action. Yet before anyone can use data as a basis for action the data have to be interpreted. The proper interpretation of data will require that the data be presented in context, and that the analysis technique used will filter out the noise."  (Donald J Wheeler, "Understanding Variation: The Key to Managing Chaos" 2nd Ed., 2000)

"Data are generally collected as a basis for action. However, unless potential signals are separated from probable noise, the actions taken may be totally inconsistent with the data. Thus, the proper use of data requires that you have simple and effective methods of analysis which will properly separate potential signals from probable noise." (Donald J Wheeler, "Understanding Variation: The Key to Managing Chaos" 2nd Ed., 2000)

On Entropy (From Fiction to Science-Fiction)

"One thinks one’s something unique and wonderful at the center of the universe. But in fact one’s merely a slight delay in the ongoing march of entropy." (Aldous Huxley, "Island", 1962)

"No structure, even an artificial one, enjoys the process of entropy. It is the ultimate fate of everything, and everything resists it." (Philip K Dick, "Galactic Pot-Healer", 1969)

"When things don't change any longer, that's the end result of entropy, the heat-death of the universe. The more things go on moving, interrelating, conflicting, changing, the less balance there is - and the more life." (Ursula K Le Guin, "The Lathe of Heaven", 1971)

"In the wastes of nonbeing it is born, flickers out, is born again and holds together, swells and spreads. In lifelessness it lives, against the gray tide of entropy it strives, improbably persists, gathering itself into ever richer complexities until it grows as a swelling wave. (James Tiptree Jr., "SheWaits for All Men Born", 1976)

"Her dance spoke of nothing more and nothing less than the tragedy of being alive, and being human. It spoke, most eloquently, of pain. It spoke, most knowingly, of despair. It spoke of the cruel humor of limitless ambition yoked to limited ability, of eternal hope invested in an ephemeral lifetime, of the driving need to try and create an inexorably predetermined future. It spoke of fear, and of hunger, and, most clearly, of the basic loneliness and alienation of the human animal. It described the universe through the eyes of man: a hostile environment, the embodiment of entropy, into which we are all thrown alone, forbidden by our nature to touch another mind save secondhand, by proxy. It spoke of the blind perversity which forces man to strive hugely for a peace which, once attained, becomes boredom. And it spoke of folly, of the terrible paradox by which man is simultaneously capable of reason and unreason, forever unable to cooperate even with himself." Spider Robinson and Jeanne Robinson, "Stardance", 1977)

"We see the universe as it is, Father Damien, and these naked truths are cruel ones. We who believe in life, and treasure it, will die. Afterward there will be nothing, eternal emptiness, blackness, nonexistence. In our living there has been no purpose, no poetry, no meaning. Nor do our deaths possess these qualities. When we are gone, the universe will not long remember us, and shortly it will be as if we had never lived at all. Our worlds and our universe will not long outlive us. Ultimately entropy will consume all, and our puny efforts cannot stay that awful end." (George R R Martin, "The Way of Cross and Dragon", 1979)

"But no longer were they always obedient to the mandates of their creators; like all material things, they were not immune to the corruptions of Time and its patient, unsleeping servant, Entropy." (Arthur C Clark, "3001: The Final Odyssey", 1997)

"Out of twinkling stardust all came, into dark matter all will fall. Death mocks us as we laugh defiance at entropy, yet ignorance birthed mortals sail forth upon time’s cruel sea." (Peter F Hamilton, "The Temporal Void", 2008)

"Yet, in the end, entropy will always emerge victorious, snuffing out the very last glimmer of heat and light. After that there is only darkness. When that state is reached even eternity will cease to exist, for one moment will be like every other and nothingness will claim the universe." (Peter F Hamilton, "The Temporal Void", 2008)

"Nothing up there tonight but entropy, and the same imaginary shapes that people had been imposing on nature since they’d first thought to wonder at the heavens." (Peter Watts, "Echopraxia", 2014)

"The process of thinking itself requires us to view the universe in the direction of entropy, since an abstraction always involves information loss, since symbols 'abstract' complexity from observed objects." (John C Wright, "Awake in the Night Land", 2014)

"And don’t ever make the mistake of thinking that things you didn’t intend or plan don’t matter. It’s a big, disorganised multiverse out there - an accident of stars. Almost nothing ever works out like we want it to, and when it does, there’s guaranteed to be unexpected consequences. Randomness is what separates life from entropy, but it’s also what makes it fun." (Foz Meadows, "An Accident of Stars", 2016)

"Entropy is just a fancy way of saying: things fall apart." (Dan Brown, "Origin", 2017)